# Solved – Continuity (from above) theorem

Let $$mathcal{A}_1 supseteq mathcal{A}_2 supseteq mathcal{A}_3 supseteq cdots$$ be a sequence of non-increasing sets with elements in the sample space $$mathcal{S}$$. Define the union of the sequence as the set:

$$mathcal{A} equiv bigcap_{k=1}^infty mathcal{A}_k.$$

Show that $$mathbb{P}(mathcal{A}) = lim_{k rightarrow infty} mathbb{P}(mathcal{A}_k)$$. Can someone please provide the proof and explanation as to why this is the case? I am confused on how I would show this to be true.

Contents

Since $$mathcal{A}_1 supseteq mathcal{A}_2 supseteq mathcal{A}_3 supseteq cdots$$ we have $$overline{mathcal{A}}_1 subseteq overline{mathcal{A}}_2 subseteq overline{mathcal{A}}_3 subseteq …$$, which means we can form disjoint sets of the form $$overline{mathcal{A}}_k – overline{mathcal{A}}_{k-1}$$ for $$k geqslant 2$$. Using DeMorgan's law and the countable additivity property, we have:
begin{equation} begin{aligned} mathbb{P}(mathcal{A}) = mathbb{P} Bigg( bigcap_{k=1}^infty mathcal{A}_{k} Bigg) &= 1 – mathbb{P} Bigg( overline{bigcap_{k=1}^infty mathcal{A}_{k}} Bigg) \[6pt] &= 1 – mathbb{P} Bigg( bigcup_{k=1}^infty overline{mathcal{A}}_k Bigg) \[6pt] &= 1 – mathbb{P} Bigg( overline{mathcal{A}}_1 cup bigcup_{k=2}^infty (overline{mathcal{A}}_k – overline{mathcal{A}}_{k-1}) Bigg) \[6pt] &= 1 – mathbb{P} ( overline{mathcal{A}}_1) + sum_{k=2}^infty mathbb{P} (overline{mathcal{A}}_k – overline{mathcal{A}}_{k-1}) \[6pt] &= 1 – mathbb{P} ( overline{mathcal{A}}_1) + sum_{k=2}^infty [ mathbb{P} (overline{mathcal{A}}_k) – mathbb{P}(overline{mathcal{A}}_{k-1}) ] \[6pt] &= 1 – mathbb{P} ( overline{mathcal{A}}_1) + lim_{n rightarrow infty} sum_{k=2}^n [ mathbb{P} (overline{mathcal{A}}_k) – mathbb{P}(overline{mathcal{A}}_{k-1}) ] \[6pt] &= lim_{n rightarrow infty} Bigg( 1 – mathbb{P} ( overline{mathcal{A}}_1) + sum_{k=2}^n [ mathbb{P} (overline{mathcal{A}}_k) – mathbb{P}(overline{mathcal{A}}_{k-1}) ] Bigg) \[6pt] &= lim_{n rightarrow infty} Big( 1 – mathbb{P} ( overline{mathcal{A}}_n) Big) \[6pt] &= lim_{n rightarrow infty} mathbb{P} ( mathcal{A}_n). \[6pt] end{aligned} end{equation}